what did GRIGORI PERELMAN discover?

everything is a sphere. unless it has holes.

[heart] scott.

also, i think he also discovered this is true in other dimensions.

but does that mean if it has a cut-out pyramid, cube, etc., it's still a sphere?!

The way I understand it, he didn't invent the "everything is a sphere" thing, that's what Poincare invented. Perelman ironed out some of the seeming problems with this concept that no one else had been able to deal with. But I could be totally misunderstanding the situation.

well it was just a conjecture, he just proved that it was always true. i would have been able to explain the rest of it a year ago but now i totally forgot what all this stuff means :(

is always true i mean, but whatever

Okay, first off I have never really studied topology, so this is all off the top of my head and its probably all wrong. Any actual mathematics person should step in now....

Perelman proved the Poincáre Conjecture, which basically says that if you have a 3-dimensional topological manifold where all anything in it that looks like a hole can be tightened down to a single point, you can actually have a 3-sphere.

So, what does that mean? First a 3-sphere: a circle is a 1-sphere, a ball is a 2-sphere, and a 3-sphere is the thing you get when you add yet another dimension. You can't visualize it, but its an equally valid mathematical entity.

A topological manifold? A manifold is a sort of space (space like you are used to) except lacking some of the restrictions we have in normal day-to-day space). Topological manifolds are just like normal day-to-day space when you look at them up close, but they may be more complicated at a distance. This is sort of analogous to looking at the earth around you and saying "looks flat from here", when really the earth is round. Manifolds are the bread and butter of topology.

So what does the Poincáre Conjecture say? It says that in this higher dimensional case, a whole bunch of different manifolds are equivalent to spheres. Equivalent means that you can start with one, push and tweak it in non-destructive ways, and end up with the other.

Why is this interesting? Really its not, especially for everybody that has been reading about topology for the first time in the new york times this week. But its a hard problem that has been studied for a century without a solution, and while people have proved this sort of thing true in lower dimensions (the ones we can visualize) and higher dimensions, but until now, not this one. It likely has all sorts of applications for other unsolved math problems, but it basically comes down to "difficult unsolved math problem getting proved is intrinsically interesting".

I think part of the confusion over this is that the Poincáre Conjecture talks about 3D manifolds. This is sortakinda like talking about a 4 dimensional object, its not the 3D that we are used to talking about. Manifolds that can be visualized in 3D space are 2D manifolds.

yeah the banach-tarski paradox is way better guys

if you wanna talk about balls

can we talk about the hairy ball theorem too?

balls balls balls

haha that was my favorite

thank you jacob! 3D mainfolds..in 4D spaces...